Path Invariance and the Robustness of Flow Matching: Beyond Architectural and Data Perturbations

Recent advances in Flow Matching (FM) models have demonstrated remarkable empirical success, often attributed to their ability to learn stable velocity fields that transport probability distributions from noise to data. Recent work has further revealed a striking form of global stability: FM models trained under severe perturbations—such as architectural changes or training on disjoint subsets of data—can still generate visually similar samples when initialized from the same noise realization. While these observations suggest that FM models learn a robust latent-to-data geometry, the precise origin of this stability remains unclear. In this work, we investigate whether this stability arises from the specific probability path used during training, potentially through a mechanism that ``locks'' trajectories onto the data manifold. To test this hypothesis, we systematically vary the probability path while fixing the initial latent seed. We compare Linear Optimal Transport (OT), Variance Preserving (VP), Linear VP, and Cosine paths on the MNIST dataset. Our results show that trajectories initialized from the same noise converge to perceptually similar samples across different paths, allowing us to rule out the hypothesis that stability is primarily induced by a particular interpolant. At the same time, we observe clear differences in optimization behavior across paths. OT-based paths converge reliably and stably, consistent with recent empirical practice, whereas Linear VP paths exhibit pronounced instability during training. These findings suggest a nuanced conclusion: while the final generative mapping appears largely invariant to the choice of probability path, the path remains crucial from the perspective of optimization and generalization. Our results indicate that Flow Matching learns a robust global geometry of the data manifold, while the probability path mainly determines how easily this geometry can be learned by a neural network.